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4
votes
1answer
69 views

A distinguished triangle of mapping spectra arising from recollement

I suspect the following should be well known, in some circles, under some name. Alas, I could not figure out how to prove it or where to look it up. Recall that a recollement is a sequence of ...
12
votes
1answer
219 views

From relative categories to marked simplicial sets

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two ...
13
votes
2answers
364 views

When do colimits agree with homotopy colimits?

I'm wondering about when the colimit and the homotopy colimit agree with diagrams of simplicial sets. I know that hocolim$(F)=$colim$(F_c)$ where $F_c$ is the cofibrant replacement of $F$. However, it ...
8
votes
1answer
224 views

Relative version of Quillen's theorem A

Quillen's Theorem A is formulated as follows: Let $F:X\to Y$ be a functor between small categories. Suppose for each $y\in Y$ the category $F/y$ is contractible. Then $F$ induces a weak equivalence ...
2
votes
2answers
93 views

Reedy model structure on sSet

According to this question, there is a model structure on $\mathrm{Set}$ in which the cofibrations are the monomorphisms, the fibrations are maps which are either epimorphisms or have empty domain, ...
15
votes
1answer
588 views

Is there an ∞-categorical interpretation of the Quillen S⁻¹S construction?

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an ...
3
votes
0answers
189 views

N-periodic derived categories

I have some seemingly basic questions about $N$-periodic derived categories to which I have not found answers in any of the usual places. Let $R$ be a ring, and let $D(R)_{\mathbb Z/N\mathbb Z}$ ...
8
votes
0answers
125 views

Model category of cofibrant topological spaces

By browsing the list of open problems in Mark Hovey's book "Model categories", I came across the following one (Problem 8.3): find a model structure on topological spaces, with the same weak ...
6
votes
1answer
171 views

Bousfield Localization and Quillen Equivalence

The notion of a (left, say) Bousfield localization of a model category doesn't seem to be invariant under Quillen equivalence. There are a lot of things that could go wrong. But I don't know any ...
4
votes
1answer
149 views

When does the projective model structure on functors exist?

What this boils down to is: if $\mathcal{K}$ is cofibrantly-generated model category which permits the small object argument and $\mathcal{D}$ is a small category, then when does ...
6
votes
0answers
349 views

Does abstract nonsense of model categories determine the “nonlinear” morphisms of $L_\infty$ algebras?

Recall that a Lie algebra is a module for the operad $\mathrm{Lie}$, which is freely generated by a binary antisymmetric operation $\beta$ modulo an equation that is quadratic in $\beta$. There is a ...
29
votes
5answers
3k views

What is modern algebraic topology(homotopy theory) about?

At a basic level, algebraic topology is the study of topological spaces by means of algebraic invariants. The key word here is "topological spaces". (Basic) algebraic topology is very useful in other ...
6
votes
4answers
637 views

A reference for Calculus of Functors for Model Categories

I am wondering where I might look to see what has been done in terms of Calculus of Functors for more general weak equivalences and Model Categories. I am at least aware of some of the extended ...
3
votes
2answers
195 views

How to simplify the proof of right-properness?

Question. Is it true that to check that a model category is right proper, it suffices to check the property for weak equivalences with fibrant codomain ? (if the domain is also fibrant, the pullback ...
5
votes
0answers
138 views

Continuous cohomology via model category

Is it possible to formulate notion of continuous cohomology in terms of model categories? If yes, then is there a reference for this?
8
votes
1answer
311 views

Non-Cartesian Monoidal Model Structure on a Slice Category

Given a monoidal model category $(M,\otimes, 1)$, and a monoid therein $A$, one can take the slice model category $M_{/A}$. This category has a natural monoidal structure induced by taking fibered ...
4
votes
0answers
109 views

What structure of a monoidal simplicial model category is preserved by taking the opposite category?

Suppose we have $(M,\otimes,1)$, a monoidal simplicial model category. Then we can consider the opposite model category $M^{op}$ with the opposite model structure (fibrations become cofibrations, ...
7
votes
1answer
194 views

Operads and the Stable Module Category

I am seeking references to places where operads and their algebras have been studied for the stable module category. Colored operads are fine too. Let $k$ be a field and $R$ a $k$-algebra. The stable ...
2
votes
1answer
74 views

Homotopy limits of homotopically constant diagrams over contractible categories

I suspect that the following result should be true and more or less well known: Let $\mathcal{M}$ be a model category and $I$ a small category with contractible nerve. For every diagram $X: I \to ...
3
votes
1answer
232 views

Reference for t-structures on stable model categories

What kind of definitions of t-structures on stable model categories have been investigated in the literature? Of course, one can always define a t-structure on a stable model category as a ...
3
votes
0answers
127 views

Test categories applied to Dold-Kan correspondence?

Let's see how this goes, this might be a bit rushed, if you spot any mistakes feel free to correct them. A test category $X$ is a category that can be used in place of the simplex category $\Delta$ to ...
9
votes
2answers
180 views

Simple question: different definitions of Bousfield localization

I am not an expert on model categories and I am getting lost with two different definitions I have found on Bousfield localizations. I don't see the link between them. First definition: Let ...
10
votes
1answer
509 views

Categorical or simplicial introduction to modern homotopy theory

I've heard some people saying that it would be easier to study homotopy theory first through simplicial sets. First, i thought that these people were not being serious. But it caught me wondering... ...
1
vote
1answer
99 views

dg-resolution of the polynomial algebra

I am intersted in constructing a cofibrant resolution of the commutative polynomial algebra in some number of variables in the category of dg-algebras(not necceserily commutative). The resolutions ...
42
votes
4answers
7k views

Do we still need model categories?

One modern POV on model categories is that they are presentations of $(\infty, 1)$-categories (namely, given a model category, you obtain an $\infty$-category by localizing at the category of weak ...
2
votes
0answers
70 views

How to show these class of morphisms are perfect

In Lurie's book "Higher Topos Theory", a class $\mathsf{W}$ of morphisms in a category $\mathcal{A}$ is called perfect if Every isomorphism belongs to $\mathsf{W}$. it satisfies "2 out of 3 ...
1
vote
0answers
65 views

Cofibrancy of simplicial objects [duplicate]

Let $\mathcal{C}$ be a site. Consider $sPsh(\mathcal{C})$ be the equipped with the local projective model structure. Let $C_{\bullet}$ be a cofibrant object in $\mathcal{C}$ and let $y(C_\bullet)$ be ...
11
votes
1answer
367 views

Examples of differential cohomology in cohesive $\infty$ topos

I might direct this question to Urs Schreiber directly, but just in case someone else has some interesting examples, I'll make the question public. The formulation of differential cohomology in ...
13
votes
1answer
560 views

The category theory of $(\infty, 1)$-categories

There are many proposed models for the theory of $(\infty, 1)$-categories and it has now been shown that many of these theories have Quillen-equivalent model categories, i.e. that they are equivalent ...
10
votes
1answer
317 views

Small objects vs Compact objects

Given a cocomplete category $C$, is there an example of an object which is small but not compact? I am working with the following definitions of small and compact: Given a cardinal $\kappa$ one ...
7
votes
1answer
180 views

Fiber vs homotopy fiber in model categories: simple question

I have a concrete problem with the homotopy fiber and I am getting lost with the literature. I state my question and, to avoid confusions, I state downwards the definitions I am using. Let $C$ be a ...
3
votes
2answers
389 views

transfinite composition of weak equivalences in sSet

Weak equivalences in the standard model structure on simplicial sets are allegedly closed under transfinite composition. What's a reference for that?
6
votes
2answers
329 views

Is dgCat a category or a 2-category?

Let us consider dgCat, the "collection" of all small dg-categories. In On differential graded categories and Lectures on dg categories the authors state that they form a category, i.e. dgCat has ...
11
votes
3answers
2k views

What is the “universal problem” that motivates the definition of homotopy limits/colimits (and more generally “derived” functors)?

The ordinary notions of limit and colimit are universal solutions to a problem, specifically, finding terminal/initial objects in slice/coslice categories. In the context of homotopy right Kan ...
13
votes
1answer
367 views

Fibrant-cofibrant models of Eilenberg-MacLane spectra

There are many models for spectra, by which I mean a model category whose homotopy category is triangulated-equivalent to the stable homotopy category. In each model, there are ways to construct ...
3
votes
0answers
119 views

Reedy fibrant and cofibrant objects

Let $\mathcal{C}$ be a Reedy category, $\mathcal{M}$ be a model category. Then consider $Fun(\mathcal{C}, \mathcal{M})$, the category of diagrams from $\mathcal{C}$ to $\mathcal{M}$ equipped with the ...
5
votes
1answer
61 views

Characterization of right properness using slice categories

I would like to know how to cite this theorem (which has a quite surprising consequence): A model category $\mathcal{M}$ is right proper if and only if for any weak equivalence $f:A\to B$, the ...
3
votes
1answer
164 views

Triangulated structure on $\mathbf{SH}(S)$: $\mathbb{P}^1$-suspension versus classical suspension

I am studying the construction of the motivic stable homotopy category of schemes $\mathbf{SH}(S)$ following Riou's paper Categorie homotopiquement stable d'un site suspendu avec intervalle (click to ...
5
votes
1answer
91 views

Injective model structure on sheaves of bounded complexes of $A$-modules

The following might be very well known for people who works with model categories, but I do not find the answer. Let $A$-be a ring. Denote $\mathbf{Ch}_+(A)$ the category of positive degree cochain ...
9
votes
1answer
299 views

Why do the model structures on dg-algebras and on dg-categories are not compatible?

First we talk about dg-algebras. According to this n-lab page, we write $dgAlg$ for the category of cochain dg-algebras in non-negative degree over a field $k$ of characteristic $0$. Write ...
2
votes
1answer
95 views

Model structures on diagrams indexed by a Reedy category

I'm interested in the way to put a model structure on the category of functors $F : P^{op} \rightarrow Ch(\mathbf{k})$ where $\mathbf{k}$ is a field of characteristic zero, $Ch(\mathbf{k})$ the ...
5
votes
1answer
283 views

Homotopy theory of acyclic categories

Homotopy theory of category of posets is well-developed and explained in various places. My interest is in acyclic categories. Recall that in acyclic categories only invertible morphisms are the ...
13
votes
1answer
395 views

Homotopy transfer in the opposite direction

Let $X\rightleftarrows Y\circlearrowleft$ be a strong deformation retraction of chain complexes (a.k.a. contraction), i.e. $X\rightarrow Y\rightarrow X$ is the identity, $Y\rightarrow Y$ is a homotopy ...
8
votes
0answers
348 views

Standard model structures on $Top$

Call a model structure on $Top$ (the category of topological spaces) standard, if the weak equivalences are the weak homotopy equivalences. In this nLab page, two standard model structures on $Top$ ...
11
votes
2answers
267 views

Quasicategories for non-simplicial model categories

If I have a simplicially enriched model category, then I can take the coherent nerve of the full subcategory of bifibrant opjects to obtain a quasicategory. If I have a model category that is not ...
7
votes
3answers
583 views

A fibrant-objects structure on Top

(Sorry for the crossposting, but I'm really interested in this question). One can define (Paragraph 1.5, page 10) a fibrant-object structure on a suitable cartesian closed category of topological ...
6
votes
2answers
506 views

On combinatorial and cellular model categories and infinity categories

I am looking for a counterexample. Let me first give the set-up. When you work with model categories, it is extremely common to assume they are cofibrantly generated. For me, this means the definition ...
8
votes
3answers
630 views

When is the projective model structure cartesian? When is the internal hom invariant?

If M is a sufficiently nice model category and D is a small category then there are two natural model structures we can impose on the functor category $Fun(D,M)$ where the weak equivalences are the ...
7
votes
1answer
290 views

Reference for a generalization of Γ-spaces to monoidal model categories

Γ-spaces were introduced by Segal in 1969 as models for what can be now described as commutative ∞-monoids and ∞-groups in cartesian symmetric monoidal ∞-categories, e.g., E_∞-spaces and connective ...
10
votes
2answers
568 views

Is there a notion of a “model category which admits left Bousfield localization?”

At a conference not too long ago I gave a talk on (left) Bousfield localization and was asked an interesting question afterwards. The question was whether I knew any examples of model categories which ...